# The meaning of 'coupling'?

In quantum mechanics if two quantities $A$ and $B$ are said to be coupled what does this actually mean?

I would guess that it means we have a term like $A\cdot B$ in the Hamiltonian but this is only a guess.

• What do you mean "two quantities"? Can you give an example other than the spin-spin or spin-orbit (i.e. spin-someotherformofangularmomentum) couplings? – ACuriousMind Jun 27 '16 at 11:49
• Usage taken from which reference? – Qmechanic Jun 27 '16 at 11:53
• @ACuriousMind (apart from orbit-obrit coupling) no, but I guess my point is why does spin-spin coupling (etc) use the word coupling, it must have some extra meaning. – Quantum spaghettification Jun 27 '16 at 11:55
• It means that the two spins that are "coupled" are not independent of each other? I'm not sure what "extra meaning" you're looking for – ACuriousMind Jun 27 '16 at 11:57
• @ACuriousMind Possibly just that expressed more mathematically, and more generally. – Quantum spaghettification Jun 27 '16 at 11:58

Let me preface by saying that "coupling" is a favorite physicist word that is perhaps best described linguistically than rigorously; it's deployed in a few different situations.

In general, we say that a coupling exists in quantum mechanics if the evolution of one part of the system depends on another quantity, which could be either classical or quantum. I'll give one example for each.

Suppose the Hamiltonian of a two-level system is an internal Hamiltonian $H_\mathrm{int}$ and an additional part that depends on some external parameter, maybe $\theta$: $$H = H_\mathrm{int} + \theta \sigma_z$$ Here the $\sigma_z$ was arbitrary. The point is that this system's evolution depends directly on the parameter $\theta$--maybe it's an external magnetic field, or some other feature of the environment. In this case, we would generally say that the system is "coupled to $\theta$." (You'd often see this in a metrological context, where we might be interested in using a quantum system coupled to an external parameter to measure the parameter.) In this case, there is only one quantum object, evolving under $H$.

Another common system--maybe a little more general--would be the evolution of two different variables both treated quantum mechanically. The idea here is that there would be some operator $A$ characterizing one observable of interest, and another $B$ characterizing a second. Then the Hamiltonian might be: $$H = H_A + H_B + H_{AB}$$ Where $H_A$ doesn't contain any term depending on $B$, $H_B$ doesn't contain any term depending on $A$, and $H_{AB}$ might have terms like $A \cdot B$, $A^2 B$, etc. The reason why this couples the system is that if we now evaluate Heisenberg equations of motion $\dot{A} = \frac{i}{\hbar} \left[ H, A \right]$ we'll find that the $H_{AB}$ term will put terms depending on $B$ into $\dot{A}$ and vice versa. Therefore, solving the equations of motion will require describing both $A$ and $B$. On the other hand, if the equations "decouple" or we do something to decouple them ourselves, we can usually find a solution for $A(t)$ that doesn't depend on $B$ and vice versa.

This is all paralleled in classical mechanics, by the way, where we would call two variables coupled if they appeared in each others' equations of motion.

EDIT: Peter Shor points out that objects can be "indirectly coupled," which is correct but would usually require me to introduce another variable $C$. I think the most general statement of being coupled/uncoupled is asking whether the equations of motion can be solved independently of each other.

• Great answer. Can you give some easy examples of a H_{AB} term in physics please? Like all what I can think about is things that are a bit advanced like the higgs field coupled to a fermoinic field (for symmetry breaking), or a higgs field coupled to R (to gravity). I would like to have some more basic examples please. – Guess601 Dec 12 '20 at 16:33
• @Guess601 a simple example might be in the Jaynes-Cummings Hamiltonian, where there is a term for an atom, a term for the electromagnetic field, and a coupling term between them. – zeldredge Dec 12 '20 at 20:40
• @zelderedge thanks for the example. I have a related question and I would really appreciate your feedback about it. It's posted in here: physics.stackexchange.com/questions/600160/… – Guess601 Dec 12 '20 at 21:03

I think Entanglement may answer your question. Two systems are said to be entangled(coupled) if we cannot assign an independent and separate wavefunctions for each system, instead we define a composite system which is simply the tensor product of the original constitutes. To be precise, in the general case the wavefunction description of any quantum system is not satisfactory, so we use a more compact form to define the general state of any quantum system and this will be the density matrix which is simply a bijective map that has the power to define both quantum populations and coherences which is something that is lacking in the wavefunction treatment, also density matrices is most suitable when dealing with open quantum systems problems.

So now you can assign a density matrix for each of your variables and another one for the composite of them both which is simply a tensor product and to say that they are coupled (entangled) is like saying that the matrix of the composite system cannot be written as the product of the original matrices.

Any interaction between two systems within the context of open quantum systems will have some washed up correlation upon tracing out any of the degrees of freedom of any of the two systems, resulting in an entagled state.

• Two quantum mechanical systems can be entangled without being coupled, and they can be coupled without being entangled. Look up "coupled harmonic oscillators". It doesn't mean "entangled harmonic oscillators". – Peter Shor Jun 27 '16 at 12:55
• Harmonic oscillators in QM can be used as a mathematical tool to quantize photon modes, to have a coupled mode equation simply means that you have a system of coupled differential equation , for example the decay of a single atom in a cavity, the dynamics of such system is governed by a coupled population equation but at the same time the state of that single atom is coupled with the state of the spontaneously emitted photon where the state of whole system will be an entangled state. So, entanglement can be be synonymous to coupling. – Elmo Jun 27 '16 at 13:08
• Two things can be entangled without being directly coupled—this is what makes entanglement so surprising and non-intuitive. And while coupled harmonic oscillators in a pure state are very likely to be entangled, if they're in the thermal state they are not entangled. – Peter Shor Jun 27 '16 at 13:12
• Definitely wrong! Entanglement is just a correlation that is preserved after an interaction that took place in the past. The main generator of entanglement in many of quantum information protocols is Spontaneous down conversion where you pump a non-linear material with high intensity photons, then due to the nonlinear interaction in the crystal these photons will decompose to two entangled photons satisfying phase matching conditions. So nothing happens without interactions – Elmo Jun 27 '16 at 13:17
• And now, you've added another paragraph with an incorrect statement about "washed-up correlation", making your answer even worse. Did you mean "measuring" rather than "tracing out"? – Peter Shor Jun 27 '16 at 20:52